# Dynamics of Phase Synchronization between Solar Polar Magnetic Fields Assessed with Van Der Pol and Kuramoto Models

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## Abstract

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## 1. Introduction

`aa`inferred from the Kuramoto and Van der Pol models of the coupling oscillators [25]. These reconstructions are built on the assumption that the coefficients of the corresponding differential equations vary slowly; in other words, the properties of the solutions are inherited from the equations with the constant coefficients. Now we address the question regarding the dynamics of the coupling and the phase difference when estimating the proximity of the reconstructions.

## 2. Data

## 3. Method: Reconstruction of the Coupling with Two Models

#### 3.1. Kuramoto Model

#### 3.2. Van Der Pol Model

#### 3.3. Reconstruction Scheme

- (A)
- Given time series $X\left(t\right)$, $Y\left(t\right)$, and the model parameter $\Delta \omega $, we reconstruct the coupling with both models. These series exhibit the solar cycle; ${T}^{*}=11$ years is used in the paper to assign a single number to the variable length of the cycle. The time axis is initially stretched by ${T}^{*}/2\pi $ to transform the estimate of the cycle length into $2\pi $ and set the correspondence between the time axis in the models and observations. Clearly, the linear transform does not affect either the correlation between the series $X\left(t\right)$ and $Y\left(t\right)$ or the variability of the solar cycle. The Kuramoto reconstruction is performed with (8) and denoted ${\mu}_{k}\left(t\right)$. The VdP reconstruction is performed with (12) and denoted ${\mu}_{v}\left(t\right)$. These two procedures are schematically displayed in the left two blocks of Figure 3. The both reconstructions ${\mu}_{k}\left(t\right)$ and ${\mu}_{v}\left(t\right)$, in general, depend on time, since the input series represent the observations instead of the solutions of the model equations. The mathematical expectation of the input series is switched into the mean when the correlation is computed.
- (B)
- Following Equation (7), we put$$\theta \left(t\right)=arcsin\frac{\Delta \omega}{\mu \left(t\right)},\phantom{\rule{1.em}{0ex}}x=sin(t+\theta \left(t\right)/2),\phantom{\rule{1.em}{0ex}}y=sin(t-\theta \left(t\right)/2)$$
- (C)
- Finally, we repeat the reconstruction of the coupling from the time series and the phase difference (the first block from the right in Figure 3). Equation (8) is applied for the both types of the input to get ${\widehat{\mu}}_{k}$ and ${\widehat{\mu}}_{v}$ from ${\rho}_{k}$ and ${\rho}_{v}$ respectively; $\Delta \omega $ has been fixed during the steps (A)–(C). This part involves the dynamics of the equations into the reconstruction. Namely, the addressed question is how the dynamics of the coupling in the direct problem affects reconstruction. We end up with the reverse transform of the time axis and restore years as the units of the reconstructions found in the paper and displayed on the Figures.

#### 3.4. Comparison of the Reconstructions

## 4. Results

#### 4.1. Reconstructed Couplings and Relation between Two Models

#### 4.2. Reconstruction of the Frequencies

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The relationship between coupling $\mu $, $\Delta \omega $, and the correlation $\rho $ between two coupled van der Pol oscillators; $\Delta \omega <\mu $.

**Figure 4.**

**Top panel**: Reconstructed coupling with the Kuramoto (${\widehat{\mu}}_{k}\left(t\right)$; green) and van der Pol (${\widehat{\mu}}_{v}\left(t\right)$; orange) models found with $\Delta \omega =0.2$.

**Bottom panel**: ratio ${r}_{\Delta \omega}[{\widehat{\mu}}_{k},{\widehat{\mu}}_{v}]\left(t\right)$ defined by (14) exhibiting the proximity of the reconstructions. Grey figure background means that the reconstruction of the coupling involves either negative correlation between the series (in the middle) or the computation of the correlation on smaller windows (at the left and right); $\Delta \omega =0.1,\phantom{\rule{0.166667em}{0ex}}0.2,\phantom{\rule{0.166667em}{0ex}}0.3$.

**Figure 5.**The minimal (blue) and maximal (red curve) values of ${r}_{\Delta \omega}\left(t\right)$ obtained with different $\Delta \omega $ vs. time.

**Figure 6.**Reconstructed normalized natural frequency, $1+\Delta \omega $, associated with the signals coming from the northern solar hemisphere. The reconstruction rule is inferred from the assumption that the Kuramoto and VdP reconstruction of the coupling result in the time-independent values of ${r}_{\Delta \omega}\left(t\right)$ fixed to $0.86$.

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**MDPI and ACS Style**

Savostianov, A.; Shapoval, A.; Shnirman, M. Dynamics of Phase Synchronization between Solar Polar Magnetic Fields Assessed with Van Der Pol and Kuramoto Models. *Entropy* **2020**, *22*, 945.
https://doi.org/10.3390/e22090945

**AMA Style**

Savostianov A, Shapoval A, Shnirman M. Dynamics of Phase Synchronization between Solar Polar Magnetic Fields Assessed with Van Der Pol and Kuramoto Models. *Entropy*. 2020; 22(9):945.
https://doi.org/10.3390/e22090945

**Chicago/Turabian Style**

Savostianov, Anton, Alexander Shapoval, and Mikhail Shnirman. 2020. "Dynamics of Phase Synchronization between Solar Polar Magnetic Fields Assessed with Van Der Pol and Kuramoto Models" *Entropy* 22, no. 9: 945.
https://doi.org/10.3390/e22090945